In a statement, the bank said the scale of the backlog was "unprecedented".
It warned that it could be a week before operations return to normal.
A computer software failure meant tens of thousands of customers could not access their accounts and wages were not paid in.
Branches opened at the weekend to deal with problems created by the failure.
Chris Sullivan, chief executive of corporate banking at RBS, which owns Ulster Bank, said every effort was being made to ensure customers were not disadvantaged.
"We are putting all of our efforts into ensuring that nobody is out of pocket in this situation. We have a team working on all eventualities," he said.
"We are really sorry this has occurred. This is the last thing any bank would want. We are taking this really seriously."
The chief executive of the Citizens Advice Bureaux (CAB), Derek Alcorn, said he has spoken to the Ulster Bank and was assured that if customers brought proof of identity into the bank, they would receive their benefits or salary.